Find a unit vector perpendicular to the plane ABC, where the coordinates of A, B and C are A(3, –1, 2), B(1, –1, –3) and C(4, –3, 1).

Given points A(3, –1, 2), B(1, –1, –3) and C(4, –3, 1)


Let position vectors of the points A, B and C be, and respectively.



We know position vector of a point (x, y, z) is given by, where, and are unit vectors along X, Y and Z directions.




Similarly, we have and


Plane ABC contains the two vectors and.


So, a vector perpendicular to this plane is also perpendicular to both of these vectors.


Recall the vector is given by







Similarly, the vector is given by







We need to find a unit vector perpendicular to and.


Recall a vector that is perpendicular to two vectors and is



Here, we have (a1, a2, a3) = (–2, 0, –5) and (b1, b2, b3) = (1, –2, –1)






Let the unit vector in the direction of be.


We know unit vector in the direction of a vector is given by .



Recall the magnitude of the vector is



Now, we find.





So, we have



Thus, the required unit vector that is perpendicular to plane ABC is.


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